Let $A,B$ be two left $R$-modules. I was wondering if we then can form the tensor product of $A$ and $B$ by the free abelian group on $A \times B$ divided out by the span of the following elements $(a+a',b) -(a,b) - (a',b), (a,b+b')-(a,b)-(a,b')$ and $(ra,b)-(a,rb)$.
In literature the construction is done for $A$ a right $R$ module and $B$ a left $R$ module. I don't see why we should distinguish between those two types.